Appendix 1 Dynamic Polarizability of an Atom

Definition of Dynamic Polarizability

The dynamic (dipole) polarizability aðoÞ is an important spectroscopic characteristic of atoms and nanoobjects describing the response to external electromagnetic disturbance in the case that the disturbing field strength is much less than the atomic

ꢀ

electric field strength E<<E_a ¼ m_e² e⁵ ꢀh⁴ ffi 5:14 Á 10⁹ V=cm, and the electromagnetic wave length is much more than the atom size.
From the mathematical point of view dynamic polarizability in the general case is the tensor of the second order a_ij connecting the dipole moment d induced in the electron core of a particle and the strength of the external electric field E (at the frequency o):

X

• d_iðoÞ ¼
• a_ijðoÞ E_jðoÞ:

(A.1)

j

For spherically symmetrical systems the polarizability a_ij is reduced to a scalar:

a_ijðoÞ ¼ aðoÞ d_ij; (A.2)

where d_ij is the Kronnecker symbol equal to one if the indices have the same values and to zero if not. Then the Eq. A.1 takes the simple form:

dðoÞ ¼ aðoÞ EðoÞ:

(A.3)
The polarizability of atoms defines the dielectric permittivity of a medium eðoÞ according to the Clausius-Mossotti equation:

eðoÞ À 1

43

¼

p n_a aðoÞ;

(A.4)

347

eðoÞ þ 2

V. Astapenko, Polarization Bremsstrahlung on Atoms, Plasmas, Nanostructures

and Solids, Springer Series on Atomic, Optical, and Plasma Physics 72,

#

• DOI 10.1007/978-3-642-34082-6,
• Springer-Verlag Berlin Heidelberg 2013

• 348
• Appendix 1 Dynamic Polarizability of an Atom

where n_a is the concentration of substance atoms. For simplicity it is assumed in Eq. A.4 that a medium consists of atoms of the same kind.
It should be noted that the basis for a number of experimental procedures of determination of the dynamic polarizability aðoÞ is its connection with the refractive index of a substance (that for a transparent nonmagnetic medium is

pffiffiffiffiffiffiffiffiffi

determined by the equation nðoÞ ¼ eðoÞ).
Dynamic polarizability defines the shift of the atomic level energy DE_n in an external electric field. In the second order of the perturbation theory for the nonresonant external field E and a spherically symmetric electronic state a corresponding correction to energy looks like

1

DE^ð_n^2Þ ¼ À a_nðoÞ E²:

(A.5)
2

The formula (A.5) describes the quadratic Stark effect. In the case that the external field frequency coincides with the eigenfrequency of an atom, the energy shift is found to be linear in electric field intensity – the linear Stark effect. The linear Stark effect is realized also in case of an orbitally degenerate atomic state as it takes place for a hydrogen atom and hydrogen-like ions.
Static polarizability, that is, polarizability at the zero frequencyaðo ¼ 0Þdefines the level shift in a constant electric field and, besides, the interatomic interaction potential at long distances (the Van der Waals interaction potential). Since static polarizability is a positive value (see below), from the Eq. A.5 it follows that the energy of a nondegenerate atomic state decreases in the presence of a constant electric field. The potential of interaction of a neutral atom with a slow charged particle at long distances is also defined by its static polarizability:

að0Þ

V_polðrÞ ¼ Àe²₀

;

(A.6)
2 r⁴

where e₀ is the particle charge. With the use of Eq. A.6 it is possible to obtain the following expression for the cross-section of elastic collision of a charged particle with an atom in case of applicability of the classical approximation for description of motion of an incident particle with the energy E:

rffiffiffiffiffiffiffiffiffi

að0Þ

el scat

s

ðEÞ ¼ 2 p e₀

:

(A.7)
2 E

It should be noted that the Eq. A.7 follows (accurate to the factor equal to 2) from
Eq. A.6 if the effective scattering radius r_E is determined with the use of the equation

ꢁꢁ
ꢁꢁ

E ¼ V_polðr_EÞ :

(A.8)

• Appendix 1 Dynamic Polarizability of an Atom
• 349

Thus the knowledge of dynamic polarizability is very important for description of a whole number of elementary processes.

Expression for the Dynamic Polarizability of an Atom

Let us calculate the dipole moment of an atom d in the monochromatic field EðtÞ ¼ 2RefE_o expðÀi o tÞg that by definition is

dðtÞ ¼ 2RefaðoÞ E_o expðÀi o tÞg:

The Fourier component of the dipole moment is given by the expression

d_o ¼ aðoÞ E_o:

(A.9)
(A.10)
In the formulas (A.9) and (A.10) E_o is the complex electric field vector in monochromatic radiation being a Fourier component of EðtÞ.
The dipole moment of an atom in the absence of external fields is equal to zero in view of spherical symmetry, so the value of an induced dipole moment can serve as a measure of disturbance of an atom by an external action. The linear dependence dðtÞon electric field intensity (A.9) is true in case of smallness of the field strengthE from the standpoint of fulfilment of the inequations E<<E_a. Thus for low enough field strengths the response of an atom to electromagnetic disturbance can be characterized by its polarizability aðoÞ.
For description of the electromagnetic response of an atom – a quantum system – within the framework of classical physics, it is convenient to use the spectroscopic conformity principle. It can be formulated as follows: an atom in interaction with an electromagnetic field behaves as a set of classical oscillators (transition oscillators) with eigenfrequencies equal to the frequencies of transitions between atomic energy levels. This means that each transition between the atomic states jji and jni is assigned an oscillator with the eigenfrequency o_jn and the damping constant d_jn< <o_jn. The contribution of the transition oscillators to the response of an atom to electromagnetic action is proportional to a dimensionless quantity called oscillator strength – f_jn , the more is the oscillator strength, the stronger is a corresponding transition. Transitions with the oscillator strength equal to zero are called forbidden transitions.
According to the spectroscopic conformity principle, the change of an atomic state is made up of the change of motion of transition oscillators. The inequation E<<E_a means the smallness of perturbation of an atomic electron state as a result of action of the electromagnetic field. Thus it is possible to consider the deviations of the transition oscillators from the equilibrium position under the action of the field EðtÞ small, so for a nth oscillator the equation of motion in the harmonic approximation is true:

• 350
• Appendix 1 Dynamic Polarizability of an Atom

e

2

• €
• _

r_n þ d_n0 r_n þ o_n0 r_n ¼ f_n0 EðtÞ;

(A.11)

m

wherer_n is the radius vector corresponding to the deviation of the transition oscillator from the equilibrium position; d_n0, o_n0, f_n0 are the damping constant, eigenfrequency and the oscillator strength. For simplicity we consider a one-electron atom in the ground state, the dipole moment of which is equal to d ¼ e r . (In case of a multielectron atom the dipole moment is equal to the sum of dipole moments of atomic electrons.) In view of the conformity principle the induced dipole moment of an atom is made up of induced dipole moments of the oscillators of transitions to

• P
• P

the nth state d_n: d ¼ d_n ¼ e we have r_n. Going in this equation to Fourier components,

• n
• n

X

d_o ¼ e

r_no

;

(A.12)

n

where r_no is the Fourier transform of the radius vector of the transition oscillator deviation from the equilibrium position. The expression for this value follows from the equation of motion (A.11):

e

f_n0

r_no

¼

E_o:

(A.13)

m o²_n0 À o² À i o d_n0

Substituting the formula (A.13) in the Eq. A.12 and using the definition of polarizability (A.10), we find for it the following expression:

e²

m

f_n0

o²_n0 À o² À i o d_n0

X

aðoÞ ¼

:

(A.14)

n

Hence it follows that the dynamic polarizability of an atom represents, generally speaking, a complex value with a dimensionality of volume. The imaginary part of polarizability is proportional to the damping constants of the transition oscillators. The sum on the right of the Eq. A.14 includes both summation over the discrete energy spectrum and integration with respect to the continuous energy spectrum. The imaginary part of polarizability is responsible for absorption of radiation, and the real part defines the refraction of an electromagnetic wave in a medium. The expression (A.14) describes not only a one-electron atom, but also a multielectron atom. The multielectron nature of an atom is taken into account by the fact that in definition of the oscillator strength the dipole moment of an atom is equal to the sum of dipole moments of its electrons.
From the Eq. A.14 several important limiting cases can be obtained. For example, if the frequency of the external field is equal to zero, the formula (A.14) gives the expression for the static polarizability of an atom:

• Appendix 1 Dynamic Polarizability of an Atom
• 351

e²

m

f_n0 o²_n0

X

a₀ ꢀ aðo ¼ 0Þ ¼

:

(A.15)

n

Hence it is seen that static polarizability is a real and positive value. It has a large numerical value if in the atomic spectrum there are transitions with high oscillator strength and low eigenfrequency as it is, for example, for alkaline-earth atoms.
In the opposite high-frequency limit, when ꢀh o>>I_P ( I_P is the ionization potential of atom) and the eigenfrequencies in the denominators of Eq. A.14 can be neglected, from the formula (A.14) in view of the golden rule of sums, according to which the sum of oscillator strengths is equal to the number of electrons in an atom N_a, we obtain

e² N_a m o²

a ðoÞ ¼ À

1

:

(A.16)
Hence it is seen that the high-frequency polarizability of an atom is a real and negative value that decreases quadratically with growing frequency of the external field.
If the external field frequency is close to one of eigenfrequencies of the transition oscillators, so that the resonance condition

jo À o_n0j ꢁ d_n0

(A.17) is satisfied and one resonant summand in the sum (A.14) can be retained, then from Eq. A.14 the expression for resonant polarizability follows:

• ꢂ
• ꢃ

• e²
• f_n0

a_resðoÞ ¼

:

(A.18)

2 m o_n0 o_n0 À o À i d_n0=2

In derivation of Eq. A.18 from Eq. A.14 in nonresonance combinations the distinction of the external field frequency from the transition eigenfrequency was neglected. Resonant polarizability is a complex value, the real part of which can be both positive and negative.
The Eq. A.10 determining dynamic polarizability after the Fourier transformation can be rewritten as

1

ð

dðtÞ ¼ aðtÞ Eðt À tÞ dt;

(A.19)

À1

• 352
• Appendix 1 Dynamic Polarizability of an Atom

where aðtÞ is the real time function, the Fourier transform of which is equal to the dynamic polarizability aðoÞ. The most simple expression for aðtÞ follows from the formula (A.18):

e² f_n0

• a_resðtÞ ¼
• ðÀiÞ yðtÞ expðÀi o_n0 t À d_n0 t=2Þ;

(A.20)
2 m o_n0

where yðtÞ is the Heaviside step function. The time dependence of the induced dipole moment dðtÞ coincides with the time dependence of the right side of the Eq. A.20 for the delta pulse of the field: EðtÞ ¼ E₀ dðtÞ, where dðtÞ is the Dirac delta function. In the general case the expression for bðtÞ can be obtained by replacement

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

of the frequency o_n0

!

o²_n0 À ðd_n0=2Þ and summation over all transition oscillators. It should be noted that the decrease of the eigenfrequency of oscillations in view of damping following from the said replacement is quite natural since friction (the analog of damping) reduces the rate of motion.
Concerned above was dipole polarizability that describes the response of an atom to a spatially uniform electric field. In the case that the characteristic size of the spatial nonuniformity of a field is less than the size of an atom, dipole polarizability should be replaced by the generalized polarizability of an atom aðo; qÞ depending on the momentum ꢀh q transferred as a result of interaction. The spatial scale of the field nonuniformity l is inversely proportional to the value of the wave vector l ꢂ 1=q. With the use of the generalized polarizability of an atom aðo; qÞ the formula (A.3) is modified to the form

ð

dq

DðoÞ ¼ aðo; qÞ Eðo; qÞ

;

(A.21)

3

ð2 pÞ

where Eðo; qÞ is the spatio-temporal Fourier transform of the electric field vector. For the spatially uniform field Eðo; qÞ ¼ EðoÞ dðqÞ the Eq. A.21 (in case of a spherically symmetric atomic state) goes to Eq. A.1 in view of the fact that

aðoÞ ¼ aðo; q ¼ 0Þ.

Appendix 2 Methods of Description of the Electron Core of Multielectron Atoms and Ions

Slater Approximation

For definition of the effective field and the concentration of a atomic core, for simple estimations, and in a number of applications, in which the behavior of wave functions of atomic electrons at long distances is essential, nodeless Slater functions of the following form are used:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2mþ1

ð2bÞ

P_gðrÞ ¼

r^me^{Àb r}

;

(A.22)

Gð2m þ 1Þ

whereg ¼ ðnlÞis the set of quantum numbers characterizing an electronic state, b; m are the Slater parameters. The wave functions (A.22) are normalized, have correct asymptotics at long distances. The main advantage of the functions (A.22) consists in their simplicity.
To determine the parameters b; m, Slater proposed empirical rules that for shells more than half-filled look like

Z À S_g

b_g ¼

;

(A.23)

m_g

where Z is the atomic nucleus charge, S_g is the screening constant, the values of which, together with the parameter m_g and the number of electrons N_g for different electron shells, are given in Table A.1.
For shells that are half-filled or less than half-filled, the best results are given by another rule:

.pffiffiffiffiffiffiffiffi

m ¼ half-integer value nearest to Z

2jEj; b ¼ m 2jEj=Z;

(A.24)

353

V. Astapenko, Polarization Bremsstrahlung on Atoms, Plasmas, Nanostructures

and Solids, Springer Series on Atomic, Optical, and Plasma Physics 72,

#

• DOI 10.1007/978-3-642-34082-6,
• Springer-Verlag Berlin Heidelberg 2013

354 Appendix 2 Methods of Description of the Electron Core of Multielectron Atoms and Ions Table A.1 Slater parameters of atomic shells

Shell g ¼ ðnlÞ

S_g

0.30

m_g
N_g

• 1s²
• 1
• 2

• 8
• 2(sp)⁸
• 4.15

11.25 21.15 27.75 39.15 45.75 44.05 57.65 71.15
2

• 3(sp)⁸
• 3
• 8

3d¹⁰

• 3
• 10

• 8
• 4(sp)⁸
• 3.5

3.5 4

4d¹⁰

10
8
5(sp)⁸ без 4f 4(df)²⁴ 5(sp)⁸ с 4f 5d¹⁰ c 4f
3.5 4
24
8

• 4
• 10

where E is the electronic state energy in atomic units.
With the use of the functions (A.22) the radial distribution of the electron density of an atom in the Slater approximation can be obtained as

X

rðrÞ ¼

N_g P²_gðrÞ:

(A.25) (A.26) (A.27)

g

The atomic (Slater) potential corresponding to this electron density is

z_SðrÞ

U_SðrÞ ¼ À

;

r

where z_SðrÞ is the effective charge of the core:

r

1

• ð
• ð

rðr⁰Þ

z_SðrÞ ¼ Z À rðr⁰Þdr⁰ À r

dr⁰: r⁰

r

0

It is possible to make sure that the potential of Eqs. A.26 and A.27 satisfies the electrostatic Poisson equation with the boundary conditions:

z_Sð0Þ ¼ Z; z_Sð1Þ ¼ Z_i;

(A.28) where Z_i is the charge of an ion that is equal, naturally, to zero for a neutral atom. Substituting in Eq. A.27 the formulas (A.22), (A.25) and performing integration, we find

• "
• #

2mÀ1

k

• X
• X

2m À k ð2b rÞ

2m k!

• z_SðrÞ ¼ Z À
• N_g 1 À e^À2br